ABSTRACT
The authors have precisely defined what it means for one group to be isomorphic to another, let's consider how they might use this definition in the context of groups that have more than just a few elements. Any property that isomorphic groups must share is known as an invariant, or technically, an invariant of group isomorphisms. Invariants are properties that must be preserved by any isomorphism. When they introduced the notation to denote the isomorphism relation between groups G and H, they claimed that this relation was in fact an equivalence relation on the set of all groups. They have already learned a few things about isomorphism classes of groups of certain orders.
